Six Sigma / SPC

Six Sigma is a development on SPC which attempts to broaden the usage
of traditional SPC (Statistical Process Control) and incorporate elements
of TQM, but both Six Sigma and SPC can be viewed as another TQM tool.

Using Excel to Calculate Sigma

Be sure to include the Equals (=) sign. This will give you your process
sigma (or sigma capability) assuming the 1.5 sigma shift.

For example if you type this into Excel,

=NORMSINV(1-100/1000000)+1.5

you will get 5.22 for your Process Sigma.

How To Determine Sample Size, Determining
Sample Size

In order to prove that a process has been improved, you must measure
the process capability before and after improvements are implemented.
This allows you to quantify the process improvement (e.g. defect reduction
or productivity increase) and translate the effects into an estimated
financial result something business leaders can understand and appreciate.
If data is not readily available for the process, how many members of
the population should be selected to ensure that the population is properly
represented? If data has been collected, how do you determine if you have
enough data?

Determining sample size is a very important issue because samples that
are too large may waste time, resources and money, while samples that
are too small may lead to inaccurate results. In many cases, we can easily
determine the minimum sample size needed to estimate a process parameter,
such as the population mean

When sample data is collected and the sample mean
is calculated, that sample mean is typically different from the population
mean
This difference between the sample and population
means can be thought of as an error. The margin of error
is the maximum difference between the observed sample mean
and the true value of the population mean
:

where:

is known as the critical value, the positive
value that is at the vertical boundary for the area of
in the right tail of the standard normal distribution.

is the population standard deviation.

is the sample size.

Rearranging this formula, we can solve for the sample size necessary
to produce results accurate to a specified confidence and margin of error.

This formula can be used when you know
and want to
determine the sample size necessary to establish, with a confidence of
, the mean value
to within
. You can still use this formula if you don’t know your population
standard deviation
and you have a small sample size. Although
it’s unlikely that you know
when the population mean
is not known, you may be able to determine
from a similar process or from a pilot test/simulation.

Example Problem
We would like to start an Internet Service Provider (ISP) and need to estimate
the average Internet usage of households in one week for our business
plan and model. How many households must we randomly select to be 95%
sure that the sample mean is within 1 minute of the population mean
. Assume that a previous survey of household usage has shown
= 6.95 minutes.

Solution
We are solving for the sample size
.

A 95% degree confidence corresponds to
= 0.05. Each
of the shaded tails in the following figure has an area of
= 0.025. The region to the left of
and
to the right of
= 0 is 0.5 - 0.025, or 0.475. In the
Table of the Standard Normal (
) Distribution, an area of 0.475 corresponds to a
value of 1.96. The critical value
is therefore
= 1.96.

The margin of error
= 1 and the standard deviation
= 6.95. Using the formula for sample size, we can calculate
:

So we will need to sample at least 186 (rounded up) randomly selected
households. With this sample we will be 95% confident that the sample
mean
will be within 1 minute of the true population of
Internet usage.

Sample Size

By definition, a sample of size n is random if the probability of selecting
the sample is the same as the probability of selecting every other sample
of size n. If the sample is not random, a bias in introduced which causes
a statistical sampling or testing error by systematically favoring some
outcomes over others. It is the responsibility of the Quality professional
to ensure that samples are random, unbiased and representative of the
population.

Let's examine three examples from manufacturing, transaction and ebusiness
life that require sampling to ensure process capability:

Parts on a manufacturing conveyor line going from
one station to the next need to be examined to ensure proper tolerancing.

Statements being stuffed into envelopes and then
sealed by an automatic machine need to be verified that they are completely
sealed.

Users visiting your Internet site and clicking
through your product catalog should be polled about their online experience.

In these three cases we would like to select a random sample of parts,
envelopes and users from a population of 1000 parts, envelopes and users
that are produced, sealed or visit the site daily. Let's assume a 95%
confidence level, 15% margin of error and population size of 1000. The
sample size needed to represent the population is 41. In each of the three
cases, there will be significant bias if we were to select the first 41
of the 1000 for that day. That would be convenience sampling and the 'early
birds' of each of the processes may not represent the population very
well. We cannot select the parts, envelopes or users that we think are
appropriate either, as this would introduce serious problems.

How do we decide which parts, envelopes and users to select for our
sampling? With a population size of 1000, we could randomly select 41
numbers between 1 and 1000. Where could we get the numbers? They could
be generated by a computer program such as Microsoft Excel. For instance,
in Excel you would use the following cell formula to derive the first
random number of the 41 needed:

=RANDBETWEEN(bottom,top)

where bottom is the smallest integer RANDBETWEEN will return (in this
case 1) and top is the largest integer RANDBETWEEN will return (in this
case 1000). If this function is not available, you may need to install
the Analysis ToolPak by selecting it the Add-Ins command on the Tools
menu.

Remember- Users visiting your Internet site always have a choice to
close the window if they prefer not to take your survey. Ensure that your
sample size is the total number of users you randomly selected minus the
number of users that refuse to provide feedback.

One final note on the sample: In the case of the parts and envelopes,
they have no choice but to be sampled if you select them. Users visiting
your site, on the other hand, always have a choice to close the window
if they prefer not to take your survey. Ensure that your sample size is
the total number of users you randomly selected minus the number of users
that refuse to provide feedback, this will give you an unbiased method
for obtaining a random sample.

Small (<30) Sample Size Calculation

The formula for the sample size necessary to produce results accurate
to a specified confidence and margin of error is:

where:

is known as the critical value, the positive
value that is at the vertical boundary for the area of
in the right tail of the standard normal distribution.

is the population standard deviation.

is the sample size.

This formula can be used when you know
and want to
determine the sample size necessary to establish, with a confidence of
, the mean value
to within
.

As a general rule of thumb, if your sample size
is greater
than 30, you can replace
by the sample standard deviation
s.

If your sample size is less than or equal to 30, the population must
be normally distributed and you must know the population standard deviation
in order to use the formula above. An alternate solution
method is to use the Student Distribution developed by William Gosset.

How to Calculate Process Sigma

Consider the power company example from the previous page: A power company
measures their performance in uptime of available power to their grid.
Here is the 5 step process to calculate your process sigma.

Step 1: Define Your Opportunities

An opportunity is the lowest defect noticeable by a customer. This definition,
of course, is debatable within the Six Sigma community. Here's a useful
snippet from the forum discussing this point:

"Typically, most products (and services) have more than one opportunity
of going wrong. For example, it is estimated than in electronics assembly
a diode could have the following opportunities for error:
1) Wrong diode and
2) wrong polarity (inserted backwards)

So for each assembly shipped, at least two defect opportunities could
be assigned for each diode. Apparently, some manufacturers of large complex
equipment with many components prefer to count two opportunities in this
case. My point is that this approach dilutes Six Sigma metrics."
-Anonymous

Many Six Sigma professionals support the counter point. The pioneer
of Six Sigma, Motorola built pagers that did not require testing prior
to shipment to the customer. Their process sigma was around six, meaning
that only approximately 3.4 pagers out of a million shipped did not function
properly when the customer received it. The customer doesn't care if the
diode is backwards or is missing, just that the pager works.

Returning to our power company example, an opportunity was defined as
a minute of up-time. That was the lowest (shortest) time period that was
noticeable by a customer.

Step 2: Define Your Defects

Defining what a defect is to your customer is not easy either. You need
to first communicate with your customer through focus groups, surveys,
or other voice of the customer tools. To Motorola pager customers, a defect
was defined as a pager that did not function properly.

In the power company example, a defect is defined by the customer as
one minute of no power. An additional defect would be noticed for every
minute that elapsed where the customer didn't have power available.

Step 3: Measure Your Opportunities and Defects

Now that you have clear definitions of what an opportunity and defect
are, you can measure them. The power company example is relatively straight
forward, but sometimes you may need to set up a formal data collection
plan and organize the process of data collection. Be sure to building
a Sound Data Collection Plan to ensure that you gather reliable and statistically
valid data.

Using the power company example, here is the data we collected:

Opportunities (last year): 525,600 minutes

Defects (last year): 500 minutes

Step 4: Calculate Your Yield

The process yield is calculated by subtracting the total number of defects
from the total number of opportunities, dividing by the total number of
opportunities, and finally multiplying the result by 100.

Step 5: Look Up Process Sigma

The final step is to look up your sigma on a sigma conversion table,
using your process yield calculated in Step 4.

Assumptions

No analysis would be complete without properly noting the assumptions
that you have made. In the above analysis, we have assumed that the standard
sigma shift of 1.5 is appropriate, the data is normally distributed, and
the process is stable. In addition, the calculations are made with using
one-tail values of the normal distribution.